Progressions | Aptitude

Progressions are fundamental sequences in mathematics where each term follows a specific pattern or rule. These structured number arrangements appear in various mathematical and real-world applications, from calculating interest rates to analyzing natural phenomena. The most common types of progressions include:

Arithmetic Progression (AP)

A sequence of numbers is called an arithmetic progression if the difference between any two consecutive terms is always the same. In simple terms, it means that the next number in the series is calculated by adding a fixed number to the previous number in the series. This fixed number is called the common difference.

For example: 2, 4, 6, 8, 10 is an AP because the difference between any two consecutive terms in the series (common difference) is the same (4 – 2 = 6 – 4 = 8 – 6 = 10 – 8 = 2).

• nth term of an AP = a + (n-1) d
• Arithmetic Mean = Sum of all terms in the AP / Number of terms in the AP
• Sum of ‘n’ terms of an AP = 0.5 n (first term + last term) = 0.5 n [ 2a + (n-1) d ]

Geometric Progression (GP)

A sequence of numbers is called a geometric progression if the ratio of any two consecutive terms is always the same. In simple terms, it means that the next number in the series is calculated by multiplying a fixed number by the previous number in the series. This fixed number is called the common ratio.

For example: 2, 4, 8, 16 is a GP because the ratio of any two consecutive terms in the series (common difference) is the same (4 / 2 = 8 / 4 = 16 / 8 = 2).

• nth term of a GP = a r^n-1
• Geometric Mean = nth root of product of n terms in the GP
• Sum of ‘n’ terms of a GP (r < 1) = [a (1 – rⁿ)] / [1 – r]
• Sum of ‘n’ terms of a GP (r > 1) = [a (rⁿ – 1)] / [r – 1]
• Sum of infinite terms of a GP (r < 1) = (a) / (1 – r)

Harmonic Progression (HP)

A sequence of numbers is called a harmonic progression if the reciprocals of the terms are in AP.

In simple terms, a, b, c, d, e, f are in HP if 1/a, 1/b, 1/c, 1/d, 1/e, 1/f are in AP.

• Harmonic Mean = (2 a b) / (a + b)

For two numbers, if A, G, and H are respectively the arithmetic, geometric, and harmonic means, then.

• A ≥ G ≥ H
• A H = G², i.e., A, G, H are in GP

Examples - Progression

Example 1:

Problem Statement: Find the nth term for the arithmetic progression (AP): 5, 9, 13, 17, …

Solution:

Here,

a = 5,
difference (d) = 9 − 5 = 4

We know that the nth term of an AP is given by: nth term = a + (n − 1)d

⇒ nth term for the given AP: = 5 + (n − 1)⋅4

⇒ nth term for the given AP: = 5 + 4n − 4

⇒ nth term for the given AP: = 4n + 1

We can verify the answer by putting values of ‘n’.

⇒ n = 1 → First term = 4⋅1 + 1 = 5
⇒ n = 2 → First term = 4⋅2 + 1 = 9
⇒ n = 3 → First term = 4⋅3 + 1 = 13
⇒ n = 4 → First term = 4⋅4 + 1 = 7

Example 2:

Problem Statement: The sum of three numbers in a GP is 36, and their product is 512. Find the numbers.

Solution:

Let the numbers be a/r, a, ar.

a/r + a + ar = 36

This simplifies to: a(1 + r + r²)/r = 36

Also, it is given that the product is 512: (a/r​) × a × (ar) = 512

⟹ a³ = 512

⟹ a = 8

Now substituting a back into the sum equation: 8(1 + r + r²) ​/ r = 36

⟹ 1 + r + r²/ r ​= 36/8​ = 9/2

Multiplying through by r: 1 + r + r² = 9/2r

⟹ 2 + 2r + 2r² = 9r​

Rearranging gives: 2r² − 7r + 2 = 0

Factoring the quadratic: (2r − 1)(r − 2) = 0

Thus, r = 1/2 ​or r = 2.

The required numbers are 4,8 and 16.

Related Articles:

• Practice Progression Aptitude Quiz Questions.
• Problems on Progressions (AP,GP, HP) | Set-2.